Looking at the references in the linked paper, this paper itself does not do a credible job of answering the question that it poses for itself. It is essentially using only correlations with national level statistics and evaluations that are riddled with myriad confounds that are likely to be more important.
The "signal" of causes flowing from lay participation is likely to be drowned out by all other kinds of "noise" that is likely to be more important with respect to every single thing one would like to see as a dependent variable.
The problem is not, as the paper claims, a lack of a good survey of lay participation in the judicial system. Instead, the problem is a lack of a good way to evaluate the causal connection of lay participation to outcomes of any kind.
Methodologies that would be more convincing, although also very difficult to apply, would be "difference within difference" or a "regression discontinuity" designs that attempt to utilize some sort of natural quasi-experiment. These methods show how there was a difference in dependent variables connected to a sudden change in the level of lay participation, perhaps at the outset of the adoption of these measures or the discontinuation of the use of these measures, or due to differences between adjacent jurisdictions that are otherwise similar, to see if there is any effect. Even then, rigorously connecting any outcome in terms of judicial independence, corruption in judiciary, output per worker, etc. would be exceedingly challenging.
For example, one might look at the differences in outcomes of English cases before and after the right to a civil jury there was profoundly narrowed, or at the differences in the outcomes of criminal trials in Japan shortly before and after it adopted lay participation in serious criminal cases not long after World War II.
In civil law countries, most lay participation is as part of a mixed lay person-judge panel in criminal cases. The level of lay participation in civil cases found in American jury trials is very rare elsewhere. Only a few countries, at most, come close, and even common law countries that use criminal jury trials use them more sparingly in most cases than the U.S. does where even traffic cases are sometimes tried to juries. The marginal degree of lay participation in most countries outside the U.S. is one reason that the effect size would be expected to be modest.
A 2003 collection of articles on the subject of lay participation in the legal system was prefaced with an article containing the following abstract:
United States scholarship on lay participation revolves around one
predominant form of lay participation, the jury (Hans & Vidmar
forthcoming 2004). However, in the legal systems of many countries,
laypeople participate as decision makers in other ways. Laypersons
serve as judges (Provine 1986), magistrates (Diamond 1993), and
private prosecutors (Perez Gil 2003). Lay and law-trained judges may
also decide cases together in mixed tribunals (Kutnjak Ivkovi6 2003;
Machura 2003; Vidmar 2002). Although diverse in structure, these
methods share with the jury a set of animating ideas about lay
involvement in legal decision making.
Many of these ideas appear to be quite compelling. But despite an
extensive body of scholarship on the functioning of the jury system,
there is limited scholarly work on how alternative methods of using
laypersons in legal decision making operate in practice. There is
even less on the political and social impact of lay participation.
Whether diverse forms of lay participation promote or undermine
democratic elements in law is a critically important yet unanswered
question. All three articles in the special issue, and the
discussant's commentary, address these important subjects, promising
to further our understanding about the benefits and the limitations of
lay involvement in the legal system.
Ultimately, this effort provided little empirical evidence. I am not aware of any high quality work that has been done to evaluate this issue empirically, other than the observation that acquittal rates are significantly different between criminal cases heard in jury trials and those heard in bench trials without a jury.
There is more work that has been done to qualitatively describe the pros and cons of different kinds of lay participation, but not much that quantifies this effect.
There is also a fairly significant literature quantifying the declining use of juries in the U.S. to resolve both criminal cases (mostly due to the rise of plea bargaining) and civil cases (due to a combination of pre-trial dispositive motion practice like motions for summary judgment, wider use of mediation, and legal culture norms more strongly favoring settlement), which has been accompanied by qualitative and anecdotal speculations on how these developments have changed the character of the U.S. legal system. See, e.g., John Quinn, "The Decline of the Civil Jury Trial: Implications for Trial Practice" Newsweek (May 18, 2022) which notes that:
According to the Administrative Office of the U.S. Courts, the rate of
federal jury trials was 5.5% in 1962 but had fallen to 1.2% by 2002.
That rate dropped to a meager 0.48% in March 2020 when the pandemic
took hold and fell further by the end of 2020. The rate of civil jury
trials in state court has similarly fallen dramatically — from around
1.8% in 1976 to between 0.6% and 0.9% as of 2019.
This report was citing Valerie Hans, et al., "The Civil Jury: Reviving an American Institution" (September 2021). See also, e.g., Albert Alschuler, "Mediation with a Mugger: The Shortage of Adjudicative Services and the Need for a Two-Tier Trial System in Civil Cases," 99 Harvard Law Review 1808 (1985).